The "contrapositive" of "if X then Y" is "if not Y then not X". One is true if and only if the other is so in order to prove "if X then Y" it is sufficient to prove "if not Y then not X".
The statement you want to prove is itself and "if and only if" statement so you really need to proce two things:
"if A= B then \(\displaystyle A\triangle B= \phi\)" and
"if \(\displaystyle A\triangle B=\phi\) then A= B"
The "contrapositive of each of those is
"if not \(\displaystyle A\triangle B= \phi\) then not A= B" and
"if not A= B then not \(\displaystyle A\triangle B= \phi\)"
Which can be stated \(\displaystyle A\ne B\) if and only if \(\displaystyle A\triangle B\ne \phi\).
Of course, \(\displaystyle A\ne B\) means that either there is an \(\displaystyle x\in A\) that is not in B or there is an \(\displaystyle x\in B\) that is not in A while \(\displaystyle A\triangle B\ne \phi\) means that there exist some \(\displaystyle x\in A\triangle B\).
So to prove this you will need to prove three different things. You will need to say "if there exist x in A that is not in B" and show "then x is not in \(\displaystyle A\triangle B\)" then say "if there exist x in B that is not in A" and show "then x is not in \(\displaystyle A\triangle B \)" then say "if there exist x in \(\displaystyle A\triangle B\)" and show "then x is in A but not in B or x is in B but not in A".
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